Willmore surfaces with nonremovable singularities and number of critical levels

Abstract

In this paper, we show that for every prescribed genus \(p\in {\mathbb {N}}\) and \(n\in {\mathbb {N}}\) with \(n\ge 3\) , there exists a closed and orientable Willmore surface in \({\mathbb {R}}^n\) of genus \(p\) with at least two nonremovable point singularities. Furthermore, we prove an energy gap theorem for smooth closed and orientable Willmore surfaces in \({\mathbb {R}}^n\) of prescribed genus \(p\in {\mathbb {N}}\) with \(n\in {\mathbb {N}}\) and \(n\ge 3\). Moreover, we show that the Willmore functional as a map on the set of smooth closed and orientable surfaces in \({\mathbb {R}}^n\) of genus \(p\in {\mathbb {N}}^*\) with \(n\in \{3, 4\}\), has only a finite number of critical levels strictly below a Douglas condition on the Willmore energy in order to exclude topological splitting.